Question

1. A synthetic fiber used in manufacturing carpet has tensile strength that is normally distributed with mean 75.5 psi and standard deviation 3.5 psi. How is the standard of the sample mean changed when the sample size is increased from n=10 to n=47 ? Round all the intermediate calculations to four decimal places.

2. Suppose X has discrete uniform distribution

f(x) = { 1/3, x=1,2,3
{ 0, otherwise

A random sample of n=35 is selected from this population. Find the probability that the sample mean is greater than 2.1 but less than 2.4. Express the final answer to four decimal places.

Answer 1

1) s=$\sigma/\sqrt n$

n=10,so

s=3.5/3.16=1.11

n=47,so

s=3.5/6.855=0.51

As the sample size is increased, standard deviation of the sample mean decreases.

2) n=35,a=1,b=3

$\mu$_X=(b+a)/2=2

$\sigma$_X= $\sqrt{((b-a+1)^2-1)/12}$ =0.8165

P(2.1<X<2.4)=P((2.1-2)/(0.8165/$\sqrt{ 35}$) )<Z<((2.4-2)/(0.8165/$\sqrt {35}$ ))=P(0.72<Z<2.9)=P(z=2.9)-P(z=0.72)=0.9981-0.7642=0.2339